Problem 35 - Algebra and Trigonometry

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$\begin{aligned} &y=\cos 2 x\\\ \end{aligned}$$

Step-by-Step Solution

Answer

The amplitude of the function \(y = \cos 2x\) is 1. The period of the function is \(\pi\).

Step 1: Find the Amplitude

The amplitude of \(y = \cos 2x\) is 1 because that is the coefficient of the cosine term, and the absolute value of 1 is 1. The amplitude of a function is the maximum distance it reaches from the centerline, or in this case, the x-axis.

Step 2: Find the Period

The period of the function \(y = \cos 2x\) can be determined by dividing \(2\pi\) by the absolute value of the coefficient of \(x\), which is 2. So, the period of the function is \(\pi\). The period of a function is how long it takes for the function to repeat its pattern.

Step 3: Graph one period of the function

To graph one period of the function, draw a cosine wave that starts at \(y = 1\) and ends at the point corresponding to one period, which is \(\pi\). The graph should also show that the function has a minimum value of \(-1\) (which is the negative of the amplitude) at \(\pi/2\), as is typical for cosine functions.

Key Concepts

Amplitude

The amplitude of a trigonometric function like the cosine or sine is the measure of its vertical stretch. It tells us the heights of the peaks and the depths of the troughs from the centerline of the graph. For the cosine function, written as \( y = a\cos(bx + c) + d \), the amplitude is \(|a|\).

• In the function \( y = \cos 2x \), there is no coefficient in front of the cosine, thus \( a = 1 \).
• So, the amplitude is \( |1| = 1 \).
• This means that the graph of \( y = \cos 2x \) oscillates from 1 to -1, centered around the x-axis (or \( y = 0 \)).

Visualizing amplitude helps us understand just how "tall" our wave is from its central line. The same insights apply to sine waves, which also demonstrate amplitude in a similar manner.

Period

The period of a trigonometric function is the interval it takes for the function to complete one full cycle of its pattern. In mathematical terms, for a function \( y = a\cos(bx + c) + d \), the period is determined by the formula \((2\pi) / |b|\).

• Looking at \( y = \cos 2x \), the value of \( b \) is 2.
• Thus, the period is \( 2\pi/2 = \pi \).
• This tells us that the cosine wave completes its cycle every \( \pi \) units along the x-axis.

A shorter period indicates that the function repeats more frequently, squeezing more cycles into a shorter x-interval. Conversely, a longer period means each cycle spans a wider interval.

Graphing Trigonometric Functions

Graphing trigonometric functions like \( y = \cos 2x \) involves plotting points over one complete period and then extending these patterns if needed. The graph begins at the maximum amplitude, \( y = 1 \), falls to the minimum \( y = -1 \) halfway through the period, and returns to the starting point by the end of the period.

• Identify key points: typically where the function hits maximum, minimum, and intercepts its axis.
• For \( \cos 2x \): start plotting at \( x = 0, y = 1 \); add the midpoint \( x = \pi/2, y = -1 \); end at \( x = \pi, y = 1 \).
• Connect these points smoothly to reveal the classic cosine wave shape.

Understanding these movements in the graphing process enhances our grasp of how transformations affect the curves. They highlight how changes in amplitude and period translate to visual adjustments in the wave patterns, ultimately clarifying the behavior of sine and cosine functions.